Transition from non-ergodic to ergodic dynamics in an autonomous discrete time crystal

Abstract

We consider an autonomous system of two coupled single-mode cavities, one of which interacts with a multimode resonator. We demonstrate that for small coupling strengths between single-mode cavities, the Loschmidt echo oscillates periodically in time and spontaneous breaking of time translation symmetry takes place. The Loschmidt echo behavior is an indication of the non-ergodic nature of the system when its evolution is time-reversible and the system retains a memory of the initial state under the action of small perturbations. This behavior reveals the presence of a time crystalline order in the autonomous system. In this regime, the system is a new class of time crystals - autonomous discrete time crystals. An increase in the coupling strength leads to a transition from periodic oscillations to an exponential decay in time of the Loschmidt echo. This corresponds to the transition from non-ergodic behavior to ergodic one in the system, and is accompanied by the disappearance of time crystalline order. We demonstrate that at the transition point the time-averaged variance of the number of photons reaches a maximum, which serves as a signature of the transition. We show that such a transition can also be observed when changing the number of degrees of freedom in the resonator, which is achieved by changing its length.